under 205 different combinations of lighting and viewing illuminations (plus 205 additional measurements for anisotropic surfaces), consisting of over 14.000 images.
A similar parameterizations is used to represent the Spatially Varying Bidirectional Reflectance Distribution Function (SVBRDF), used to describe opaque, smooth materials that can have different reflectance at each point of the surface (non-homogeneous materials) [HF13]. The SVBRDF parameterization hence must takes into account the location over the surface: SVBRDF (x, y, θi, ϕi, θr, ϕr). Capturing the SVBRDF sometimes requires long measurements and processing times as well as large, specialized and expensive hardware rigs, although under certain assumptions approximate measurements can be performed even with cellphone or tablet cameras [AWL15, RPG15]. The SVBRDF cannot describe subsurface scattering and mesoscopic effects.
For a homogeneous material that can reflect light but also transmit it through its surface we need to reintroduce the transmittance angles (θt, ϕt), thus obtaining the Bidirectional Scattering Distribution Function (BSDF), comprising scattering effects for both reflection and transmission: BSDF(θi, ϕi, θr, ϕr, θt, ϕt). The BSDF can describe both transparent and opaque materials.
If we take into account only the transmittance of homogeneous material it is possible to describe it with the Bidirectional Transmittance Distribution Function (BTDF), suitable to model how the light passes through transparent or semi-transparent surfaces [WMLT07], [HF13]: BTDF(θi, ϕi, θt, ϕt).
An opaque, smooth, homogeneous material can be represented with the Bidirectional Reflectance Distribution Function (BRDF): BRDF(θi, ϕi, θr, ϕr). By looking at Figure 2.2 it is possible to note how the BRDF can be considered a special case of the more complex functions described above [ASMS01]. In fact, the BRDF can be considered as a special case of the BTF and the SVBRDF when the position on the sample surface is fixed; any BTF datasets can be approximated as a sparse linear combination of rotated analytical BRDFs [WDR11] and the SVBRDF parameterization includes extra parameters with respect to the BRDF simply to take into account the location over the surface, but it must fulfill the BRDF reciprocity and energy conservation properties, which will be described in the next sections. Finally, a BSDF can be modeled as a sum of a BRDF (for the reflection component) and a BTDF (for the transmittance component).
2.1 DEFINITION OF THE BRDF
As discussed in the previous section, one of the possible ways to represent the way an opaque, homogeneous material interacts with the light is through the BRDF (Bidirectional Reflectance Distribution Function), a radiometric function currently used to varying levels of accuracy in photorealistic rendering systems. It describes, in the general case, how incident energy redirects in all directions across a hemisphere above the surface. Historically, the BRDF was defined and suggested over the more generalized BSSRDF (Bidirectional Scattering Surface Reflectance Distribution function) [JMLH01] by Nicodemus [NRH*77], as a simplified reflectance representation for opaque surfaces: the BRDF assumes that light entering a material leaves the material at the same position, whereas the BSSRDF can describe light transport between any two incident rays on a surface. Many common translucent materials like milk, skin and alabaster cannot be represented by a BRDF since they are characterized by their subsurface scattering behavior that smooths the surface details, with the light shining through them [GLL*04]. These materials are expensive to measure and render. However many techniques have been proposed JMLH01], [DS03], [HBV03], [DWd*08], [DI11], [KRP*15].
Before defining the reflectance we provide a brief introduction to some important radiometric terms. Radiant Energy, the basic unit of energy, is measured in Joules [J] and indicated with the symbol Q:
where h is the Planck’s constant, c is the speed of light in vacuum and λ is the wavelength of the incident photon. The energy flowing through a surface per unit time is called Radiant Flux, indicated with Φ and measured in Watts [W]:
The flux flowing per unit of surface area is called radiant flux area density, measured [W/m2] and indicated by u:
Two different terms are used in order to distinguish the flow of energy toward a surface from the flow leaving a surface: in the first case we refer to irradiance (E); in the second one the term used is radiosity (B). If instead of referring to the ratio of flux per unit of surface area we take into account a solid angle, we can define the intensity I, the radiant energy leaving a point in the direction Φ per unit solid angle, measured in [W/sr]:
Finally, the radiant flux per unit solid angle and per unit projected area is called radiance:
In the following we indicate the radiance arriving at a surface with Li and the radiance leaving a surface with Lr.
We are now ready to define the BRDF as the ratio of the reflected radiance Lr to incident irradiance E:
where vi and vr are vectors describing the incident (i) and exitant (r) directions. By taking into account the incident radiance Li instead of Ei, thus considering the solid angle around the incident lighting direction and the cosine of the angle between the latter and the surface normal, we can write the Equation (2.7) in a different form, which allows understanding how the units of a BRDF are inverse steradian [1/sr]:
Researchers have measured hundreds of BRDFs, suggested implementation techniques and utilized user input to edit and enhance materials. Recent implementations have expanded material libraries but have not improved significantly upon material representation efficiency. However, the uptake of acquired models has not been widespread across rendering packages due to their data and storage requirements.
To understand the way the BRDF is parameterized, let’s take into consideration a point p on a surface and the surface normal n at that specific location on the surface; on the plane tangent to the surface in p we fix a reference direction t, called tangent direction, and its perpendicular direction b on the plane: n × t × b defines a local reference frame. Once we set the incoming light direction and the outgoing direction (viewing direction), the angle between the surface normal and the viewing direction is called θi; similarly the angle between the surface normal and the outgoing direction is called θr. If we take the projection of the viewing direction on the tangent plane, the angles between the tangent direction and the projection of the incoming direction are called respectively ϕi, and ϕr.
Figure 2.3 shows the geometry of the BRDF and the vectors used for parameterizations:
• n is the normal at
